TY - JOUR

T1 - Pathologies of the large- N limit for RPN-1 , CPN-1 , QPN-1 and mixed isovector/isotensor σ -models

AU - Sokal, Alan D.

AU - Starinets, Andrei O.

N1 - Funding Information:
We wish to thank Paolo Butera, Sergio Caracciolo, Hervé Kunz, Andrea Pelissetto and Paolo Rossi for many helpful discussions, and Larry Glasser for correspondence. In particular, it was Paolo Butera who pointed out to one of us (A.D.S.), nearly a decade ago, that the N=∞ phase transition in RP N−1 models occurs also in dimension d=1 — a comment that directly inspired the present work. This research was supported in part by U.S. National Science Foundation grants PHY-9520978 and PHY-9900769. It was also aided by the gracious hospitality of the Scuola Normale Superiore.

PY - 2001/5/14

Y1 - 2001/5/14

N2 - We compute the phase diagram in the N→∞ limit for lattice RPN-1, CPN-1 and QPN-1 σ-models with the quartic action, and more generally for mixed isovector/isotensor models. We show that the N=∞ limit exhibits phase transitions that are forbidden for any finite N. We clarify the origin of these pathologies by examining the exact solution of the one-dimensional model: we find that there are complex zeros of the partition function that tend to the real axis as N→∞. We conjecture the correct phase diagram for finite N as a function of the spatial dimension d. Along the way, we prove some new correlation inequalities for a class of N-component σ-models, and we obtain some new results concerning the complex zeros of confluent hypergeometric functions.

AB - We compute the phase diagram in the N→∞ limit for lattice RPN-1, CPN-1 and QPN-1 σ-models with the quartic action, and more generally for mixed isovector/isotensor models. We show that the N=∞ limit exhibits phase transitions that are forbidden for any finite N. We clarify the origin of these pathologies by examining the exact solution of the one-dimensional model: we find that there are complex zeros of the partition function that tend to the real axis as N→∞. We conjecture the correct phase diagram for finite N as a function of the spatial dimension d. Along the way, we prove some new correlation inequalities for a class of N-component σ-models, and we obtain some new results concerning the complex zeros of confluent hypergeometric functions.

KW - Mixed isovector/isotensor model

KW - Nematic liquid crystal N→∞ limit 1/N expansion

KW - Nonlinear σ -model

KW - RP model CP model QP model N -vector model

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U2 - 10.1016/S0550-3213(01)00065-7

DO - 10.1016/S0550-3213(01)00065-7

M3 - Article

AN - SCOPUS:0035858799

SN - 0550-3213

VL - 601

SP - 425

EP - 502

JO - Nuclear Physics B

JF - Nuclear Physics B

IS - 3

ER -